Optimal. Leaf size=134 \[ \frac{2 \sqrt{e+f x} \sqrt{a d-b c} \sqrt{\frac{b (c+d x)}{b c-a d}} E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt{d} \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}} \]
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Rubi [A] time = 0.0399542, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {114, 113} \[ \frac{2 \sqrt{e+f x} \sqrt{a d-b c} \sqrt{\frac{b (c+d x)}{b c-a d}} E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt{d} \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}} \]
Antiderivative was successfully verified.
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Rule 114
Rule 113
Rubi steps
\begin{align*} \int \frac{\sqrt{e+f x}}{\sqrt{a+b x} \sqrt{c+d x}} \, dx &=\frac{\left (\sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{e+f x}\right ) \int \frac{\sqrt{\frac{b e}{b e-a f}+\frac{b f x}{b e-a f}}}{\sqrt{a+b x} \sqrt{\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}}} \, dx}{\sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}\\ &=\frac{2 \sqrt{-b c+a d} \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{e+f x} E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{-b c+a d}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt{d} \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}\\ \end{align*}
Mathematica [A] time = 0.742917, size = 154, normalized size = 1.15 \[ \frac{2 \sqrt{c+d x} \left (\frac{(a f-b e) \sqrt{\frac{b (e+f x)}{f (a+b x)}} E\left (\sin ^{-1}\left (\frac{\sqrt{a-\frac{b e}{f}}}{\sqrt{a+b x}}\right )|\frac{b c f-a d f}{b d e-a d f}\right )}{b \sqrt{a-\frac{b e}{f}} \sqrt{\frac{b (c+d x)}{d (a+b x)}}}+\frac{e+f x}{\sqrt{a+b x}}\right )}{d \sqrt{e+f x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 209, normalized size = 1.6 \begin{align*} -2\,{\frac{ \left ({a}^{2}df-abcf-bead+{b}^{2}ce \right ) \sqrt{dx+c}\sqrt{bx+a}\sqrt{fx+e}}{{b}^{2}d \left ( bdf{x}^{3}+adf{x}^{2}+bcf{x}^{2}+bde{x}^{2}+acfx+adex+bcex+ace \right ) }\sqrt{{\frac{d \left ( bx+a \right ) }{ad-bc}}}{\it EllipticE} \left ( \sqrt{{\frac{d \left ( bx+a \right ) }{ad-bc}}},\sqrt{{\frac{ \left ( ad-bc \right ) f}{d \left ( af-be \right ) }}} \right ) \sqrt{-{\frac{ \left ( dx+c \right ) b}{ad-bc}}}\sqrt{-{\frac{ \left ( fx+e \right ) b}{af-be}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{f x + e}}{\sqrt{b x + a} \sqrt{d x + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b x + a} \sqrt{d x + c} \sqrt{f x + e}}{b d x^{2} + a c +{\left (b c + a d\right )} x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{e + f x}}{\sqrt{a + b x} \sqrt{c + d x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{f x + e}}{\sqrt{b x + a} \sqrt{d x + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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